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Maxwell's theory of solid angle and the construction of knotted fields
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Binysh, Jack and Alexander, Gareth P. (2018) Maxwell's theory of solid angle and the construction of knotted fields. Journal of Physics A: Mathematical and Theoretical, 51 (38). 385202. doi:10.1088/1751-8121/aad8c6
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Official URL: https://doi.org/10.1088/1751-8121/aad8c6
Abstract
We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in R3. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighbourhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Research Centres > Centre for Complexity Science Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Physics |
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Library of Congress Subject Headings (LCSH): | Knot theory, Geometry | ||||||||||||
Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||||||||||
Publisher: | Institute of Physics | ||||||||||||
Official Date: | 21 August 2018 | ||||||||||||
Dates: |
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Volume: | 51 | ||||||||||||
Number: | 38 | ||||||||||||
Article Number: | 385202 | ||||||||||||
DOI: | 10.1088/1751-8121/aad8c6 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 21 August 2018 | ||||||||||||
Date of first compliant Open Access: | 21 August 2018 | ||||||||||||
RIOXX Funder/Project Grant: |
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